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Abstract

We present a scheme to measure the orbital angular momentum spectrum of light using a precisely timed optical loop and quantum non-demolition measurements. We also discuss the influence of imperfect optical components.

S0 and S1 are optical switches that can be switched from been transmittive to reflective. S0 needs to transmit the initial incident light, and be switched to be reflective by the end of the first outer-loop cycle. A high repetition rate is not required and it can be implemeted either mechanically or opto-electrically. Alternatively it could also be a static high-reflectance mirror, if the one-time transmission loss at the very beginning can be tolerated. S1 needs to be switched every QZI loop cycle (ΔT) and need to be polarization insensitive. The fastest switches are Pockels cells, which can operate at 10–100 GHz with 99% transmission. One scheme, similar to the one implemented in Ref. [20], is to an interferometer with Pockels cells in one of the arms. The Pockels cells introduce π phase shift in the arm when activated, and thus switch the beam between two output ports of the interferometer. Using two Pockels cells rotated relative to each other will cancel the birefringent effect.

Technically, each |α|2 is slightly different, but the difference is well within 1%. The |α|2 that matters most is the one corresponding to the QZI loop (including S1), which, without any optimization, consists of 4 beam splitters, 5 mirrors, 1 waveplate and up to three Pockels cells. Assuming all optics are anti-reflection coated so that loss is 1% at each Pockels cell and 0.1% at each other component, we have |α|2 = 0.96.

Other (2)

S0 and S1 are optical switches that can be switched from been transmittive to reflective. S0 needs to transmit the initial incident light, and be switched to be reflective by the end of the first outer-loop cycle. A high repetition rate is not required and it can be implemeted either mechanically or opto-electrically. Alternatively it could also be a static high-reflectance mirror, if the one-time transmission loss at the very beginning can be tolerated. S1 needs to be switched every QZI loop cycle (ΔT) and need to be polarization insensitive. The fastest switches are Pockels cells, which can operate at 10–100 GHz with 99% transmission. One scheme, similar to the one implemented in Ref. [20], is to an interferometer with Pockels cells in one of the arms. The Pockels cells introduce π phase shift in the arm when activated, and thus switch the beam between two output ports of the interferometer. Using two Pockels cells rotated relative to each other will cancel the birefringent effect.

Technically, each |α|2 is slightly different, but the difference is well within 1%. The |α|2 that matters most is the one corresponding to the QZI loop (including S1), which, without any optimization, consists of 4 beam splitters, 5 mirrors, 1 waveplate and up to three Pockels cells. Assuming all optics are anti-reflection coated so that loss is 1% at each Pockels cell and 0.1% at each other component, we have |α|2 = 0.96.

Figures (6)

A schematic of the compact OAM spectrometer. The Quantum Zeno Interrogator (shaded region) distinguishes between zero and nonzero OAM states. The outer loop decreases the OAM value of light by one per round trip. All the beam splitters are polarizing beam splitters (PBSs) that transmits horizontally polarized light and reflects vertically polarized light. The OAM filter transmits states with zero OAM, but blocks states with non-zero OAM. S0 and S1 are switching mirrors that either transmits or reflects incident light [21]. R1 and R2 are fixed polarization rotators, which can be half wave plates. P1 and P2 are fast polarization switches, such as Pockels cells. When activated, P1 and P2 switches horizontal polarization to vertical and vice versa. When de-activated, they are transparent to light. The shaded region is a Quantum Zeno Interrogator [20] which separates OAM components with l = 0 and l ≠ 0 into different polarizations. Hence at PBS3, zero OAM component is sent to the detector while the none-zero OAM component is sent back into the outer-loop. The outer loop decreased OAM by one per round trip via, for example, a vortex phase plate (VPP) [22].

The probability of detecting the correct OAM value as a function of the number of loops (N) in the QZI using a perfect OAM filter. (a) Neglect optical loss. (b) Assume |α|2 = 0.96 based on commercially available optics. When optical loss is included, there exists an optimal N for higher order OAM states, due to the compromise between the quantum Zeno enhancement and optical loss.

The probabilities of different outcomes of a QZI interrogation as a function of the transmission of the OAM filter, neglecting optical loss. The blue solid line represents detecting OAM=0, the red dashed line is detecting OAM≠ 0, and the orange dotted line, loss. (a) N = 8. (b) N = 2 – 10.

(a) Extinction ratio η as a function of the number of loops N for various losses |α|2. Solid symbols are for l0 = 1 and open symbols are for l0 = 3. l0 > 3 are essentially indistinguishable from l0 = 3. For the l0 = 0 case, the extinction ratio is over a 1000 for all |α|2 values because no premature measurements are possible. The additional green crosses labeled as |α|2 = 0.95* represents |α|2 = 0.96 but including misalignment of the OAM filter and VPP as discussed in the text. (b) Extinction ratio η as a function of the normalized aperture size a0 for l0 = 6, Δl = 1 – 3, N = 8, and |α|2 = 0.96. Skipping OAM states increases the extinction ratio by orders of magnitude.

(a) The probability of measuring an OAM value l for a given input state l0 (Eq. (2)), using pinhole as the OAM filter, N = 8, |α|2 = 0.96, and misalignment of 10% and 1%, respectively, at the pinhole filter and VPP. Despite the decrease in probability for the diagonal elements at large l0, the off diagonal elements decrease much faster, as implied by the large extinction ratios. (b) The diagonal elements of (a) as a function of N for l0 = 0 – 10.